Edexcel A-Level Maths: Mixed Past Paper Skills (Page 43) | Edexcel A-Level 数学：真题混合技能（第43页）

📚 Edexcel A-Level Maths: Mixed Past Paper Skills (Page 43) | Edexcel A-Level 数学：真题混合技能（第43页）

This revision note focuses on the mixed skills usually tested around page 43 of Edexcel A-Level Mathematics past paper compilations. The page often brings together pure topics such as algebraic manipulation, trigonometric equations, calculus and vectors. Understanding how these skills are combined is essential for scoring high marks on Paper 1 and Paper 2.

本复习讲义聚焦 Edexcel A-Level 数学真题合订本第43页常见的混合技能。该页通常综合考查纯数学主题，如代数变形、三角方程、微积分和向量。理解这些技能的交叉运用对于在 Paper 1 和 Paper 2 中取得高分至关重要。

1. Reading the Question and Allocating Marks | 读题与分值分配

Mixed practice pages often combine several Assessment Objectives. The first step is to read the stem and each part carefully. Underline command words such as ‘Show that’, ‘Hence’, or ‘Find the exact value’. The number of marks gives a strong hint about the expected working.

混合练习页通常综合考查多个评估目标。第一步是仔细阅读题干和每一小问。在 ‘Show that’、’Hence’ 或 ‘Find the exact value’ 等指令词下画线。分值强烈提示了预期步骤的多少。

For example, a ‘Show that’ question usually requires a clear chain of algebra; a ‘Hence’ part depends on the previous result; an exact value means no decimal approximation unless the question says otherwise.

例如，’Show that’ 题通常要求清晰的代数推理链；’Hence’ 小问依赖前面的结果；’exact value’ 表示除非题目另有说明，否则不能给出小数近似。

• Show that — 证明：需要完整推导出给定结果。
• Hence — 因此：必须使用上一小问的结论。
• Find the exact value — 求精确值：保留根号、π 或分数形式。
• Give your answer to 3 significant figures — 答案保留 3 位有效数字。

2. Algebraic Fractions and Simplification | 代数分式与化简

Mixed questions frequently ask you to simplify rational expressions before differentiating or solving. Factorise the numerator and denominator completely, then cancel only common factors. State any values that would make the original denominator zero.

混合题中经常要求你在微分或求解之前化简有理式。先将分子和分母完全因式分解，然后只约去公因式。要注明会使原分母为零的值。

A common error is to cancel terms rather than factors. For example, in (x + 2)/(x² + 2), x cannot be cancelled because it is a term, not a factor. Always factorise first.

一个常见错误是约去项而不是因式。例如在 (x + 2)/(x² + 2) 中，不能约去 x，因为它是项而不是因式。一定要先因式分解。

(x² − 3x + 2) / (x² − 1) = [(x − 1)(x − 2)] / [(x − 1)(x + 1)] = (x − 2) / (x + 1), x ≠ 1, −1

After simplification, any excluded values remain based on the original expression. These restrictions are important when stating domains or solving equations.

化简后，任何被排除的值仍然基于原表达式。在说明定义域或解方程时，这些限制非常重要。

3. Functions, Domains and Ranges | 函数、定义域与值域

Questions often define f(x) and g(x), then ask for fg(x), inverse functions, or the range of a function. The domain is the allowed set of input values; the range is the set of outputs the function can produce.

题目经常定义 f(x) 和

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